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          电影<<社交网络>>中的"FaceMash"算法
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<blockquote>
<p>最近在看<a target="_blank" rel="noopener" href="https://movie.douban.com/subject/3205624/">&lt;&lt;社交网络&gt;&gt;</a>时,发现了一个用于投票排名的算法,自己折腾实现了一下.</p>
</blockquote>
<p>在影片中,卷西饰演的扎克伯格在被妹子甩了之后(其实是他自己直男癌),一气之下黑了附近女生宿舍的照片数据库打算做一个<code>FaceMash</code>(通过投票的方式来选出漂亮的女生,同时它也是<code>Facebook</code>的前身,后来这个网站由于流量太大,搞崩了哈佛大学的网络而被强行关闭了),并使用了他的好基友爱德华多用于计算国际象棋排名的算法.</p>
<p><strong>这是一部很好看的电影,如果没有看过我强烈推荐去看一看.</strong></p>
<h3 id="公式"><a href="#公式" class="headerlink" title="公式"></a>公式</h3><hr>
<p><img src="http://wx4.sinaimg.cn/large/63503acbly1fhpbwf3qy2j20vx0hx74u.jpg"></p>
<p>这个算法是用来计算<code>期望胜率</code>的,但影片中其实写的是错误的,正确的公式应该为: </p>
<p>$$E_a = \frac{1} {1 + 10 ^ {(R_b - R_a) / 400}}$$</p>
<ul>
<li>$E_a$就是<code>a</code>的期望胜率.</li>
</ul>
<ul>
<li>$R_b,R_a$是<code>b</code>与<code>a</code>的<code>Rank</code>分数.</li>
</ul>
<ul>
<li>当$R_a,R_b$都相同时,它们的<code>期望胜率</code>都为<code>0.5</code>,即$E_a = \frac{1} {1+10^0} = 0.5$.</li>
</ul>
<p>电影中只给出了计算<code>期望胜率</code>的算法,但我们还需要一个计算新的<code>Rank</code>分数的算法,公式如下: </p>
<p>$$R_n = R_o + K(W - E)$$</p>
<ul>
<li>$R_n$代表新的<code>Rank</code>,$R_o$自然就是旧的<code>Rank</code>了.</li>
</ul>
<ul>
<li><code>K</code>为一个定值,我把它设为<code>10</code>.</li>
</ul>
<ul>
<li><code>W</code>是<code>胜负值</code>,胜者为<code>1</code>,败者为<code>0</code>;<code>E</code>就是我们上面计算的<code>期望胜率</code>.</li>
</ul>
<h3 id="实现"><a href="#实现" class="headerlink" title="实现"></a>实现</h3><hr>
<p>有了这两个核心公式,我们就可以开始实现这个算法了,但在代码实现之前,我们先验证一下公式: </p>
<p>假设有两个女孩<code>A</code>与<code>B</code>,她们的基础<code>Rank</code>都为<code>1400</code>,通过上述的推论我们已经得知,<strong><code>当A</code>,<code>B</code>的分值相同时,她们的期望胜率都为0.5</strong>.</p>
<p>如果,我选择了<code>A</code>,则<code>A</code>的胜负值变为<code>1</code>,<code>B</code>的胜负值为<code>0</code>,然后我们套用公式2可以得出: </p>
<ul>
<li>$R_a = 1400 + 10 * (1 - 0.5) = 1405$</li>
</ul>
<ul>
<li>$R_b = 1400 + 10 * (0 - 0.5) = 1395$</li>
</ul>
<p>由于她们的分数不再相同,所以套用公式1计算现在的<code>期望胜率</code>: </p>
<ul>
<li>$R_a = \frac{1} {1 + 10 ^ {(1395 - 1405) / 400}} \approx 0.51439 $</li>
</ul>
<ul>
<li>$R_b = \frac{1} {1 + 10 ^ {(1405 - 1395) / 400}} \approx 0.48561$</li>
</ul>
<p>下面是我用<code>C</code>写的一个小程序,它初始化了两个”女孩”,然后根据输入来判断哪个胜出,并动态计算<code>Rank</code>与<code>期望胜率</code>.</p>
<p><img src="http://wx1.sinaimg.cn/mw690/63503acbly1fhpcccyw1bj20di0gcdg7.jpg"></p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">&lt;stdio.h&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">&lt;math.h&gt;</span></span></span><br><span class="line"></span><br><span class="line"><span class="keyword">typedef</span> <span class="class"><span class="keyword">struct</span> &#123;</span></span><br><span class="line">    <span class="keyword">const</span> <span class="keyword">char</span> *name;</span><br><span class="line">    <span class="keyword">int</span> rank;</span><br><span class="line">    <span class="keyword">double</span> expect_rate;</span><br><span class="line">&#125; girl;</span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> K = <span class="number">10</span>;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">read_girl</span><span class="params">(girl g)</span> </span>&#123;</span><br><span class="line">    <span class="built_in">printf</span>(<span class="string">&quot;Girl name: %s, rank: %d, expect_rate: %.5f\n&quot;</span>,g.name,g.rank,g.expect_rate);</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">compute_expect_rate</span><span class="params">(girl *a,girl *b)</span> </span>&#123;</span><br><span class="line">    <span class="keyword">int</span> a_rank = a-&gt;rank;</span><br><span class="line">    <span class="keyword">int</span> b_rank = b-&gt;rank;</span><br><span class="line">    <span class="comment">// expect rate formula</span></span><br><span class="line">    <span class="comment">// Ea = 1 / (1 + 10 ^ ((Rb-Ra) / 400))</span></span><br><span class="line">    <span class="keyword">double</span> a_rank_differ = (<span class="keyword">double</span>) (b_rank - a_rank) / <span class="number">400</span>;</span><br><span class="line">    <span class="keyword">double</span> a_rank_rate = <span class="built_in">pow</span>(<span class="number">10</span>,a_rank_differ);</span><br><span class="line">    a-&gt;expect_rate = <span class="number">1</span> / (<span class="number">1</span> + a_rank_rate);</span><br><span class="line">    <span class="comment">// Eb = 1 / (1 + 10 ^ ((Ra-Rb) / 400))</span></span><br><span class="line">    <span class="keyword">double</span> b_rank_differ = (<span class="keyword">double</span>) (a_rank - b_rank) / <span class="number">400</span>;</span><br><span class="line">    <span class="keyword">double</span> b_rank_rate = <span class="built_in">pow</span>(<span class="number">10</span>,b_rank_differ);</span><br><span class="line">    b-&gt;expect_rate = <span class="number">1</span> / (<span class="number">1</span> + b_rank_rate);</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">// new rank formula: Rn = Ro + K(W - E)</span></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">compute_rank</span><span class="params">(girl *a,girl *b,<span class="keyword">int</span> a_win_rate,<span class="keyword">int</span> b_win_rate)</span> </span>&#123;</span><br><span class="line">    a-&gt;rank = a-&gt;rank + K * (a_win_rate - a-&gt;expect_rate);</span><br><span class="line">    b-&gt;rank = b-&gt;rank + K * (b_win_rate - b-&gt;expect_rate);</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">(<span class="keyword">int</span> argc,<span class="keyword">char</span> *argv[])</span> </span>&#123;</span><br><span class="line">    <span class="keyword">char</span> a_girl_name[<span class="number">20</span>];</span><br><span class="line">    <span class="keyword">char</span> b_girl_name[<span class="number">20</span>];</span><br><span class="line"></span><br><span class="line">    girl a = &#123;.name = <span class="string">&quot;A Gril&quot;</span>,.rank = <span class="number">1400</span>&#125;;</span><br><span class="line">    girl b = &#123;.name = <span class="string">&quot;B Gril&quot;</span>,.rank = <span class="number">1400</span>&#125;;</span><br><span class="line"></span><br><span class="line">    compute_expect_rate(&amp;a,&amp;b);</span><br><span class="line"></span><br><span class="line">    read_girl(a);</span><br><span class="line">    read_girl(b);</span><br><span class="line"></span><br><span class="line">    <span class="keyword">while</span> (<span class="number">1</span>) &#123;</span><br><span class="line">        <span class="keyword">char</span> choice[<span class="number">2</span>];</span><br><span class="line">        <span class="built_in">printf</span>(<span class="string">&quot;Choice A or B?\n&quot;</span>);</span><br><span class="line">        <span class="built_in">scanf</span>(<span class="string">&quot;%s&quot;</span>,choice);</span><br><span class="line">        <span class="keyword">if</span> (choice[<span class="number">0</span>] == <span class="string">&#x27;A&#x27;</span>) &#123;</span><br><span class="line">            compute_rank(&amp;a,&amp;b,<span class="number">1</span>,<span class="number">0</span>);</span><br><span class="line">            compute_expect_rate(&amp;a,&amp;b);</span><br><span class="line">        &#125; <span class="keyword">else</span> <span class="keyword">if</span> (choice[<span class="number">0</span>] == <span class="string">&#x27;B&#x27;</span>) &#123;</span><br><span class="line">            compute_rank(&amp;a,&amp;b,<span class="number">0</span>,<span class="number">1</span>);</span><br><span class="line">            compute_expect_rate(&amp;a,&amp;b);</span><br><span class="line">        &#125; <span class="keyword">else</span> &#123;</span><br><span class="line">            <span class="built_in">printf</span>(<span class="string">&quot;Invalid choice!\n&quot;</span>);</span><br><span class="line">            <span class="keyword">break</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        read_girl(a);</span><br><span class="line">        read_girl(b);</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
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<p>本文作者为<a target="_blank" rel="noopener" href="https://github.com/SylvanasSun">SylvanasSun(sylvanassun_xtz@163.com)</a>,转载请务必指明原文链接.</p>
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